I was just about to advise you to check which path your "external" storage directory points to

For distribution with Google Play, the OBB file will be downloaded during installation of the app, and you do not have the possibility to change the target directory. So your method is correct for the "final" app.

I don't know whether the system can be fooled such that, for your experiments, you replace Environment.getExternalStorageDirectory() with the path to your "real" SD card, and then it will try to mount the OBB file there. Apps like "ES File Explorer" might show the real path. If you try, please post the results.